Biostat/Biomath M257
System information (for reproducibility):
versioninfo()Julia Version 1.10.2
Commit bd47eca2c8a (2024-03-01 10:14 UTC)
Build Info:
Official https://julialang.org/ release
Platform Info:
OS: macOS (arm64-apple-darwin22.4.0)
CPU: 12 × Apple M2 Max
WORD_SIZE: 64
LIBM: libopenlibm
LLVM: libLLVM-15.0.7 (ORCJIT, apple-m1)
Threads: 8 default, 0 interactive, 4 GC (on 8 virtual cores)
Environment:
JULIA_NUM_THREADS = 8
JULIA_EDITOR = codeLoad packages:
using Pkg
Pkg.activate(pwd())
Pkg.instantiate()
Pkg.status() Activating project at `~/Documents/github.com/ucla-biostat-257/2024spring/slides/08-numalgintro`Status `~/Documents/github.com/ucla-biostat-257/2024spring/slides/08-numalgintro/Project.toml`
[6e4b80f9] BenchmarkTools v1.5.0
[0e44f5e4] Hwloc v3.0.1
[bdcacae8] LoopVectorization v0.12.169
[6f49c342] RCall v0.14.1
[37e2e46d] LinearAlgebra
[9a3f8284] RandomTopics in numerical algebra:
Except for the iterative methods, most of these numerical linear algebra tasks are implemented in the BLAS and LAPACK libraries. They form the building blocks of most statistical computing tasks (optimization, MCMC).
Our major goal (or learning objectives) is to
All high-level languages (Julia, Matlab, Python, R) call BLAS and LAPACK for numerical linear algebra.
BLAS stands for basic linear algebra subprograms.
See netlib for a complete list of standardized BLAS functions.
There are many implementations of BLAS.
There are 3 levels of BLAS functions.
| Level | Example Operation | Name | Dimension | Flops |
|---|---|---|---|---|
| 1 | \(\alpha \gets \mathbf{x}^T \mathbf{y}\) | dot product | \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^n\) | \(2n\) |
| 1 | \(\mathbf{y} \gets \mathbf{y} + \alpha \mathbf{x}\) | axpy | \(\alpha \in \mathbb{R}\), \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^n\) | \(2n\) |
| 2 | \(\mathbf{y} \gets \mathbf{y} + \mathbf{A} \mathbf{x}\) | gaxpy | \(\mathbf{A} \in \mathbb{R}^{m \times n}\), \(\mathbf{x} \in \mathbb{R}^n\), \(\mathbf{y} \in \mathbb{R}^m\) | \(2mn\) |
| 2 | \(\mathbf{A} \gets \mathbf{A} + \mathbf{y} \mathbf{x}^T\) | rank one update | \(\mathbf{A} \in \mathbb{R}^{m \times n}\), \(\mathbf{x} \in \mathbb{R}^n\), \(\mathbf{y} \in \mathbb{R}^m\) | \(2mn\) |
| 3 | \(\mathbf{C} \gets \mathbf{C} + \mathbf{A} \mathbf{B}\) | matrix multiplication | \(\mathbf{A} \in \mathbb{R}^{m \times p}\), \(\mathbf{B} \in \mathbb{R}^{p \times n}\), \(\mathbf{C} \in \mathbb{R}^{m \times n}\) | \(2mnp\) |
The form of a mathematical expression and the way the expression should be evaluated in actual practice may be quite different.
Some operations appear as level-3 but indeed are level-2.
Example 1. A common operation in statistics is column scaling or row scaling \[
\begin{eqnarray*}
\mathbf{A} &=& \mathbf{A} \mathbf{D} \quad \text{(column scaling)} \\
\mathbf{A} &=& \mathbf{D} \mathbf{A} \quad \text{(row scaling)},
\end{eqnarray*}
\] where \(\mathbf{D}\) is diagonal. For example, in generalized linear models (GLMs), the Fisher information matrix takes the form
\[
\mathbf{X}^T \mathbf{W} \mathbf{X},
\] where \(\mathbf{W}\) is a diagonal matrix with observation weights on diagonal.
Column and row scalings are essentially level-2 operations!
using BenchmarkTools, LinearAlgebra, Random
Random.seed!(257) # seed
n = 2000
A = rand(n, n) # n-by-n matrix
d = rand(n) # n vector
D = Diagonal(d) # diagonal matrix with d as diagonal2000×2000 Diagonal{Float64, Vector{Float64}}:
0.0416032 ⋅ ⋅ ⋅ … ⋅ ⋅ ⋅
⋅ 0.887679 ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ 0.102233 ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ 0.08407 ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ … ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋮ ⋱
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ … ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅ ⋅ ⋅ ⋅ 0.213471 ⋅ ⋅
⋅ ⋅ ⋅ ⋅ ⋅ 0.870533 ⋅
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 0.318876Dfull = diagm(d) # convert to full matrix2000×2000 Matrix{Float64}:
0.0416032 0.0 0.0 0.0 … 0.0 0.0 0.0
0.0 0.887679 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.102233 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.08407 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 … 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
⋮ ⋱
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 … 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.213471 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.870533 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.318876# this is calling BLAS routine for matrix multiplication: O(n^3) flops
# this is SLOW!
@benchmark $A * $DfullBenchmarkTools.Trial: 96 samples with 1 evaluation. Range (min … max): 51.614 ms … 57.005 ms ┊ GC (min … max): 0.00% … 2.41% Time (median): 51.760 ms ┊ GC (median): 0.00% Time (mean ± σ): 52.256 ms ± 866.879 μs ┊ GC (mean ± σ): 0.86% ± 1.22% ▃█ ▄██▅▄▁▂▁▁▁▁▁▁▁▁▁▁▁▁▂▁▁▁▁▁▁▃▅▇▅▄▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▁▂ ▁ 51.6 ms Histogram: frequency by time 54.7 ms < Memory estimate: 30.52 MiB, allocs estimate: 2.
# dispatch to special method for diagonal matrix multiplication.
# columnwise scaling: O(n^2) flops
@benchmark $A * $DBenchmarkTools.Trial: 1897 samples with 1 evaluation. Range (min … max): 2.127 ms … 3.999 ms ┊ GC (min … max): 0.00% … 22.40% Time (median): 2.442 ms ┊ GC (median): 0.00% Time (mean ± σ): 2.632 ms ± 438.140 μs ┊ GC (mean ± σ): 10.90% ± 12.67% ▆█▇▅▄ ▃▆██████▇▆▇▆▆▇▆▆▆▅▅▄▄▄▃▃▃▁▂▂▂▁▂▂▂▂▁▁▁▂▄▅██▆▇▇▆▅▆▅▅▄▄▃▄▄▃▃▃▃ ▄ 2.13 ms Histogram: frequency by time 3.51 ms < Memory estimate: 30.52 MiB, allocs estimate: 2.
# Or we can use broadcasting (with recycling)
@benchmark $A .* transpose($d)BenchmarkTools.Trial: 1883 samples with 1 evaluation. Range (min … max): 2.126 ms … 4.233 ms ┊ GC (min … max): 0.00% … 27.57% Time (median): 2.469 ms ┊ GC (median): 0.00% Time (mean ± σ): 2.652 ms ± 451.323 μs ┊ GC (mean ± σ): 10.93% ± 12.70% ▅█▆▂ ▃▆█████▇▆▆▅▅▇▆▅▅▆▆▄▃▃▃▃▂▂▂▂▂▂▂▂▂▂▁▂▄▇█▇█▆▅▄▅▄▄▄▄▃▄▄▃▂▂▂▂▁▂▂ ▄ 2.13 ms Histogram: frequency by time 3.67 ms < Memory estimate: 30.52 MiB, allocs estimate: 2.
# in-place: avoid allocate space for result
# rmul!: compute matrix-matrix product A*B, overwriting A, and return the result.
@benchmark rmul!($A, $D)BenchmarkTools.Trial: 3417 samples with 1 evaluation. Range (min … max): 1.263 ms … 4.679 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 1.460 ms ┊ GC (median): 0.00% Time (mean ± σ): 1.461 ms ± 84.548 μs ┊ GC (mean ± σ): 0.00% ± 0.00% ▂▂▄▆▄▇▆▄▄▆▃▃▄▂▁▄▂▁▄▆▄▆▆█▆▄▃ ▂▂▁▂▃▃▃▃▄▃▃▄▄▄▅▄▆▆▆▆▇█████████████████████████████▆▅▅▃▃▄▃▃ ▅ 1.26 ms Histogram: frequency by time 1.6 ms < Memory estimate: 0 bytes, allocs estimate: 0.
# In-place broadcasting
@benchmark $A .= $A .* transpose($d)BenchmarkTools.Trial: 3174 samples with 1 evaluation. Range (min … max): 1.423 ms … 5.454 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 1.567 ms ┊ GC (median): 0.00% Time (mean ± σ): 1.573 ms ± 91.401 μs ┊ GC (mean ± σ): 0.00% ± 0.00% ▂▃▃█▆▇▅█▅▄▃ ▂▁▂▁▂▂▂▁▁▂▂▂▁▂▂▂▂▂▂▂▂▂▂▂▄▅▇████████████▆▅▅▃▃▃▃▃▃▃▃▃▃▂▃▃▃▂▂ ▄ 1.42 ms Histogram: frequency by time 1.67 ms < Memory estimate: 0 bytes, allocs estimate: 0.
Exercise: Try @turbo (SIMD) and @tturbo (multi-threaded SIMD) from LoopVectorization.jl package.
Note: In R or Matlab, diag(d) will create a full matrix. Be cautious using diag function: do we really need a full diagonal matrix?
using RCall
R"""
d <- runif(5)
diag(d)
"""RObject{RealSxp}
[,1] [,2] [,3] [,4] [,5]
[1,] 0.3485362 0.0000000 0.0000000 0.0000000 0.0000000
[2,] 0.0000000 0.5413159 0.0000000 0.0000000 0.0000000
[3,] 0.0000000 0.0000000 0.1638823 0.0000000 0.0000000
[4,] 0.0000000 0.0000000 0.0000000 0.5827122 0.0000000
[5,] 0.0000000 0.0000000 0.0000000 0.0000000 0.9024851# This works only when Matlab is installed
using MATLAB
mat"""
d = rand(5, 1)
diag(d)
"""ArgumentError: ArgumentError: Package MATLAB not found in current path.
- Run `import Pkg; Pkg.add("MATLAB")` to install the MATLAB package.Example 2. Innter product between two matrices \(\mathbf{A}, \mathbf{B} \in \mathbb{R}^{m \times n}\) is often written as \[ \text{trace}(\mathbf{A}^T \mathbf{B}), \text{trace}(\mathbf{B} \mathbf{A}^T), \text{trace}(\mathbf{A} \mathbf{B}^T), \text{ or } \text{trace}(\mathbf{B}^T \mathbf{A}). \] They appear as level-3 operation (matrix multiplication with \(O(m^2n)\) or \(O(mn^2)\) flops).
Random.seed!(123)
n = 2000
A, B = randn(n, n), randn(n, n)
# slow way to evaluate tr(A'B): 2mn^2 flops
@benchmark tr(transpose($A) * $B)BenchmarkTools.Trial: 95 samples with 1 evaluation. Range (min … max): 50.924 ms … 59.158 ms ┊ GC (min … max): 0.00% … 3.94% Time (median): 52.136 ms ┊ GC (median): 0.00% Time (mean ± σ): 52.486 ms ± 1.530 ms ┊ GC (mean ± σ): 0.86% ± 1.46% █ ▅ █▅▃▄▆▃▇▅████▃▁▆▅▁▃▃▄▁▄▅▄▄▃▁▃▁▃▁▃▄▁▁▁▁▁▃▁▁▁▁▁▁▁▁▁▁▁▁▁▁▃▁▁▁▁▃ ▁ 50.9 ms Histogram: frequency by time 58.3 ms < Memory estimate: 30.52 MiB, allocs estimate: 2.
But \(\text{trace}(\mathbf{A}^T \mathbf{B}) = <\text{vec}(\mathbf{A}), \text{vec}(\mathbf{B})>\). The latter is level-1 BLAS operation with \(O(mn)\) flops.
# smarter way to evaluate tr(A'B): 2mn flops
@benchmark dot($A, $B)BenchmarkTools.Trial: 2732 samples with 1 evaluation. Range (min … max): 1.649 ms … 1.984 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 1.823 ms ┊ GC (median): 0.00% Time (mean ± σ): 1.827 ms ± 31.177 μs ┊ GC (mean ± σ): 0.00% ± 0.00% ▁▁▁▃▄▇█▇▄▄▁ ▂▁▁▁▁▁▁▁▁▁▁▁▂▁▂▁▂▁▁▁▁▁▁▁▁▂▅▆████████████▇▆▅▄▄▃▃▃▃▃▃▃▃▃▃▃▂▃ ▃ 1.65 ms Histogram: frequency by time 1.93 ms < Memory estimate: 0 bytes, allocs estimate: 0.
Example 3. Similarly \(\text{diag}(\mathbf{A}^T \mathbf{B})\) can be calculated in \(O(mn)\) flops.
# slow way to evaluate diag(A'B): O(n^3)
@benchmark diag(transpose($A) * $B)BenchmarkTools.Trial: 93 samples with 1 evaluation. Range (min … max): 50.993 ms … 100.033 ms ┊ GC (min … max): 0.00% … 1.47% Time (median): 52.402 ms ┊ GC (median): 0.00% Time (mean ± σ): 53.965 ms ± 6.802 ms ┊ GC (mean ± σ): 0.90% ± 1.25% █▅▇▅▄ █████▅▅▅▁▅▁▁▁▁▇▅▁▁▁▁▁▁▁▁▅▁▁▁▁▁▁▁▁▅▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▅ ▁ 51 ms Histogram: log(frequency) by time 88.9 ms < Memory estimate: 30.53 MiB, allocs estimate: 3.
# smarter way to evaluate diag(A'B): O(n^2)
@benchmark Diagonal(vec(sum($A .* $B, dims = 1)))BenchmarkTools.Trial: 1294 samples with 1 evaluation. Range (min … max): 3.305 ms … 5.589 ms ┊ GC (min … max): 0.00% … 23.28% Time (median): 3.625 ms ┊ GC (median): 0.00% Time (mean ± σ): 3.860 ms ± 513.706 μs ┊ GC (mean ± σ): 8.86% ± 10.65% ▄█▆▃ ▆████████▇▇▆▇█▄█▅▅▃▄▂▂▂▂▂▁▁▁▁▁▁▁▁▁▁▁▂▁▂▄▇▇▇▇▇▅▆▆▅▅▄▅▄▄▃▃▃▂▂ ▄ 3.3 ms Histogram: frequency by time 4.88 ms < Memory estimate: 30.53 MiB, allocs estimate: 5.
To get rid of allocation of intermediate arrays at all, we can just write a double loop or use dot function.
function diag_matmul!(d, A, B)
m, n = size(A)
@assert size(B) == (m, n) "A and B should have same size"
fill!(d, 0)
for j in 1:n, i in 1:m
d[j] += A[i, j] * B[i, j]
end
Diagonal(d)
end
d = zeros(eltype(A), size(A, 2))
@benchmark diag_matmul!($d, $A, $B)BenchmarkTools.Trial: 1479 samples with 1 evaluation. Range (min … max): 3.117 ms … 3.586 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 3.372 ms ┊ GC (median): 0.00% Time (mean ± σ): 3.379 ms ± 41.213 μs ┊ GC (mean ± σ): 0.00% ± 0.00% ▁ ▁ ▆▅█▇▆▅▂ ▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▂▁▁▁▁▁▁▁▆█▇██████████▇▆▆▄▄▄▄▄▄▄▃▄▃▃▃ ▃ 3.12 ms Histogram: frequency by time 3.5 ms < Memory estimate: 0 bytes, allocs estimate: 0.
Exercise: Try @turbo (SIMD) and @tturbo (multi-threaded SIMD) from LoopVectorization.jl package.
Key to high performance is effective use of memory hierarchy. True on all architectures.
Source: https://cs.brown.edu/courses/csci1310/2020/assign/labs/lab4.html
Hwloc.jl package. For example, this laptop runs an Apple M2 Max chip with 4 efficiency cores and 8 performance cores.using Hwloc
topology_graphical()/------------------------------------------------------------------------------------------------------------------------------------------------------------------\
| Machine (2208MB total) |
| |
| /----------------------------------------------------------------------------------------------------------------------------------------\ /------------------\ |
| | Package L#0 | | CoProc opencl0d0 | |
| | | | | |
| | /------------------------------------------------------------------------------------------------------------------------------------\ | | 38 compute units | |
| | | NUMANode L#0 P#0 (2208MB) | | | | |
| | \------------------------------------------------------------------------------------------------------------------------------------/ | | 72 GB | |
| | | \------------------/ |
| | /----------------------------------------------------------------\ /----------------------------------------------------------------\ | |
| | | L2 (4096KB) | | L2 (16MB) | | |
| | \----------------------------------------------------------------/ \----------------------------------------------------------------/ | |
| | | |
| | /-------------\ /-------------\ /-------------\ /-------------\ /-------------\ /-------------\ /-------------\ /-------------\ | |
| | | L1d (64KB) | | L1d (64KB) | | L1d (64KB) | | L1d (64KB) | | L1d (128KB) | | L1d (128KB) | | L1d (128KB) | | L1d (128KB) | | |
| | \-------------/ \-------------/ \-------------/ \-------------/ \-------------/ \-------------/ \-------------/ \-------------/ | |
| | | |
| | /-------------\ /-------------\ /-------------\ /-------------\ /-------------\ /-------------\ /-------------\ /-------------\ | |
| | | L1i (128KB) | | L1i (128KB) | | L1i (128KB) | | L1i (128KB) | | L1i (192KB) | | L1i (192KB) | | L1i (192KB) | | L1i (192KB) | | |
| | \-------------/ \-------------/ \-------------/ \-------------/ \-------------/ \-------------/ \-------------/ \-------------/ | |
| | | |
| | /-------------\ /-------------\ /-------------\ /-------------\ /-------------\ /-------------\ /-------------\ /-------------\ | |
| | | Core L#0 | | Core L#1 | | Core L#2 | | Core L#3 | | Core L#4 | | Core L#5 | | Core L#6 | | Core L#7 | | |
| | | | | | | | | | | | | | | | | | | |
| | | /---------\ | | /---------\ | | /---------\ | | /---------\ | | /---------\ | | /---------\ | | /---------\ | | /---------\ | | |
| | | | PU L#0 | | | | PU L#1 | | | | PU L#2 | | | | PU L#3 | | | | PU L#4 | | | | PU L#5 | | | | PU L#6 | | | | PU L#7 | | | |
| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
| | | | P#0 | | | | P#1 | | | | P#2 | | | | P#3 | | | | P#4 | | | | P#5 | | | | P#6 | | | | P#7 | | | |
| | | \---------/ | | \---------/ | | \---------/ | | \---------/ | | \---------/ | | \---------/ | | \---------/ | | \---------/ | | |
| | \-------------/ \-------------/ \-------------/ \-------------/ \-------------/ \-------------/ \-------------/ \-------------/ | |
| | | |
| | /----------------------------------------------------------------\ | |
| | | L2 (16MB) | | |
| | \----------------------------------------------------------------/ | |
| | | |
| | /-------------\ /-------------\ /-------------\ /-------------\ | |
| | | L1d (128KB) | | L1d (128KB) | | L1d (128KB) | | L1d (128KB) | | |
| | \-------------/ \-------------/ \-------------/ \-------------/ | |
| | | |
| | /-------------\ /-------------\ /-------------\ /-------------\ | |
| | | L1i (192KB) | | L1i (192KB) | | L1i (192KB) | | L1i (192KB) | | |
| | \-------------/ \-------------/ \-------------/ \-------------/ | |
| | | |
| | /-------------\ /-------------\ /-------------\ /-------------\ | |
| | | Core L#8 | | Core L#9 | | Core L#10 | | Core L#11 | | |
| | | | | | | | | | | |
| | | /---------\ | | /---------\ | | /---------\ | | /---------\ | | |
| | | | PU L#8 | | | | PU L#9 | | | | PU L#10 | | | | PU L#11 | | | |
| | | | | | | | | | | | | | | | | | | |
| | | | P#8 | | | | P#9 | | | | P#10 | | | | P#11 | | | |
| | | \---------/ | | \---------/ | | \---------/ | | \---------/ | | |
| | \-------------/ \-------------/ \-------------/ \-------------/ | |
| \----------------------------------------------------------------------------------------------------------------------------------------/ |
\------------------------------------------------------------------------------------------------------------------------------------------------------------------/For example, Xeon X5650 CPU has a theoretical throughput of 128 DP GFLOPS but a max memory bandwidth of 32GB/s.
Can we keep CPU cores busy with enough deliveries of matrix data and ship the results to memory fast enough to avoid backlog?
Answer: use high-level BLAS as much as possible.
| BLAS | Dimension | Mem. Refs. | Flops | Ratio |
|---|---|---|---|---|
| Level 1: \(\mathbf{y} \gets \mathbf{y} + \alpha \mathbf{x}\) | \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^n\) | \(3n\) | \(2n\) | 3:2 |
| Level 2: \(\mathbf{y} \gets \mathbf{y} + \mathbf{A} \mathbf{x}\) | \(\mathbf{x}, \mathbf{y} \in \mathbb{R}^n\), \(\mathbf{A} \in \mathbb{R}^{n \times n}\) | \(n^2\) | \(2n^2\) | 1:2 |
| Level 3: \(\mathbf{C} \gets \mathbf{C} + \mathbf{A} \mathbf{B}\) | \(\mathbf{A}, \mathbf{B}, \mathbf{C} \in\mathbb{R}^{n \times n}\) | \(4n^2\) | \(2n^3\) | 2:n |
Source: Jack Dongarra’s slides.
A distinction between LAPACK and LINPACK (older version of R uses LINPACK) is that LAPACK makes use of higher level BLAS as much as possible (usually by smart partitioning) to increase the so-called level-3 fraction.
To appreciate the efforts in an optimized BLAS implementation such as OpenBLAS (evolved from GotoBLAS), see the Quora question, especially the video. Bottomline is
Get familiar with (good implementations of) BLAS/LAPACK and use them as much as possible.
Data layout in memory affects algorithmic efficiency too. It is much faster to move chunks of data in memory than retrieving/writing scattered data.
Storage mode: column-major (Fortran, Matlab, R, Julia) vs row-major (C/C++).
Cache line is the minimum amount of cache which can be loaded and stored to memory.
# Apple Silicon (M1/M2 chips) don't have L3 cache
Hwloc.cachelinesize()ErrorException: Your system doesn't seem to have an L3 cache.Accessing column-major stored matrix by rows (\(ij\) looping) causes lots of cache misses.
Take matrix multiplication as an example \[ \mathbf{C} \gets \mathbf{C} + \mathbf{A} \mathbf{B}, \quad \mathbf{A} \in \mathbb{R}^{m \times p}, \mathbf{B} \in \mathbb{R}^{p \times n}, \mathbf{C} \in \mathbb{R}^{m \times n}. \] Assume the storage is column-major, such as in Julia. There are 6 variants of the algorithms according to the order in the triple loops.
jki or kji looping:# inner most loop
for i in 1:m
C[i, j] = C[i, j] + A[i, k] * B[k, j]
end- `ikj` or `kij` looping:# inner most loop
for j in 1:n
C[i, j] = C[i, j] + A[i, k] * B[k, j]
endijk or jik looping:# inner most loop
for k in 1:p
C[i, j] = C[i, j] + A[i, k] * B[k, j]
end| Variant | A Stride | B Stride | C Stride |
|---|---|---|---|
| \(jki\) or \(kji\) | Unit | 0 | Unit |
| \(ikj\) or \(kij\) | 0 | Non-Unit | Non-Unit |
| \(ijk\) or \(jik\) | Non-Unit | Unit | 0 |
Apparently the variants \(jki\) or \(kji\) are preferred.
"""
matmul_by_loop!(A, B, C, order)
Overwrite `C` by `A * B`. `order` indicates the looping order for triple loop.
"""
function matmul_by_loop!(A::Matrix, B::Matrix, C::Matrix, order::String)
m = size(A, 1)
p = size(A, 2)
n = size(B, 2)
fill!(C, 0)
if order == "jki"
@inbounds for j = 1:n, k = 1:p, i = 1:m
C[i, j] += A[i, k] * B[k, j]
end
end
if order == "kji"
@inbounds for k = 1:p, j = 1:n, i = 1:m
C[i, j] += A[i, k] * B[k, j]
end
end
if order == "ikj"
@inbounds for i = 1:m, k = 1:p, j = 1:n
C[i, j] += A[i, k] * B[k, j]
end
end
if order == "kij"
@inbounds for k = 1:p, i = 1:m, j = 1:n
C[i, j] += A[i, k] * B[k, j]
end
end
if order == "ijk"
@inbounds for i = 1:m, j = 1:n, k = 1:p
C[i, j] += A[i, k] * B[k, j]
end
end
if order == "jik"
@inbounds for j = 1:n, i = 1:m, k = 1:p
C[i, j] += A[i, k] * B[k, j]
end
end
end
using Random
Random.seed!(123)
m, p, n = 2000, 100, 2000
A = rand(m, p)
B = rand(p, n)
C = zeros(m, n);using BenchmarkTools
@benchmark matmul_by_loop!($A, $B, $C, "jki")BenchmarkTools.Trial: 86 samples with 1 evaluation. Range (min … max): 58.056 ms … 59.729 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 58.433 ms ┊ GC (median): 0.00% Time (mean ± σ): 58.453 ms ± 234.579 μs ┊ GC (mean ± σ): 0.00% ± 0.00% ▁ ▁█▂▁▇ ▅▁▂ ▃▁▁▃▆▁▅▃▆█▆█████████▅▅▁▅▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▃▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▃ ▁ 58.1 ms Histogram: frequency by time 59.6 ms < Memory estimate: 0 bytes, allocs estimate: 0.
@benchmark matmul_by_loop!($A, $B, $C, "kji")BenchmarkTools.Trial: 27 samples with 1 evaluation. Range (min … max): 185.771 ms … 190.199 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 186.726 ms ┊ GC (median): 0.00% Time (mean ± σ): 187.133 ms ± 1.237 ms ┊ GC (mean ± σ): 0.00% ± 0.00% ▃█ █ ▃ ▃ ▇▇▇▁██▇▁█▁▁▁▇▇█▇▁▇▁▁▁▁▁▁▁▁▁▁▁█▇▁▇▁▁▁▇▁▁▁▁▇▁▁▇▁▁▁▁▁▁▁▁▇▁▁▁▁▁▁▇ ▁ 186 ms Histogram: frequency by time 190 ms < Memory estimate: 0 bytes, allocs estimate: 0.
@benchmark matmul_by_loop!($A, $B, $C, "ikj")BenchmarkTools.Trial: 10 samples with 1 evaluation. Range (min … max): 512.687 ms … 522.756 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 515.130 ms ┊ GC (median): 0.00% Time (mean ± σ): 515.765 ms ± 2.925 ms ┊ GC (mean ± σ): 0.00% ± 0.00% ▁▁ ▁▁▁ ▁ █ ▁ ▁ ██▁▁▁▁▁▁▁▁███▁▁▁▁█▁▁█▁▁▁▁▁▁▁▁▁▁▁▁█▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁█ ▁ 513 ms Histogram: frequency by time 523 ms < Memory estimate: 0 bytes, allocs estimate: 0.
@benchmark matmul_by_loop!($A, $B, $C, "kij")BenchmarkTools.Trial: 10 samples with 1 evaluation. Range (min … max): 508.427 ms … 513.273 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 509.480 ms ┊ GC (median): 0.00% Time (mean ± σ): 510.326 ms ± 2.027 ms ┊ GC (mean ± σ): 0.00% ± 0.00% ▁ █▁ ▁ ▁ ▁ ▁▁ ▁ █▁██▁▁▁█▁▁▁▁▁▁▁▁▁▁█▁▁▁▁▁▁▁▁█▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁██▁█ ▁ 508 ms Histogram: frequency by time 513 ms < Memory estimate: 0 bytes, allocs estimate: 0.
@benchmark matmul_by_loop!($A, $B, $C, "ijk")BenchmarkTools.Trial: 20 samples with 1 evaluation. Range (min … max): 247.799 ms … 257.115 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 249.820 ms ┊ GC (median): 0.00% Time (mean ± σ): 250.799 ms ± 2.507 ms ┊ GC (mean ± σ): 0.00% ± 0.00% ▁ ▁ ▁▁▁█▁█ ▁█ ▁ ▁ █ ▁ ▁ ▁ █▁█▁▁▁▁██████▁██▁█▁▁▁▁█▁▁▁▁▁▁▁▁▁▁▁█▁▁▁▁▁▁▁█▁▁▁▁▁▁▁█▁▁▁▁▁▁▁▁▁█ ▁ 248 ms Histogram: frequency by time 257 ms < Memory estimate: 0 bytes, allocs estimate: 0.
@benchmark matmul_by_loop!($A, $B, $C, "ijk")BenchmarkTools.Trial: 21 samples with 1 evaluation. Range (min … max): 246.205 ms … 256.408 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 249.057 ms ┊ GC (median): 0.00% Time (mean ± σ): 249.382 ms ± 2.296 ms ┊ GC (mean ± σ): 0.00% ± 0.00% ▃ ▃ ▃▃ █ ▇▁▇▁▁▇▁█▁▁▇▁▁▁█▁██▁▇▇█▁▁▁▁▁▁▁▇▁▁▁▇▁▁▁▁▁▁▇▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▇ ▁ 246 ms Histogram: frequency by time 256 ms < Memory estimate: 0 bytes, allocs estimate: 0.
@benchmark mul!($C, $A, $B)BenchmarkTools.Trial: 1882 samples with 1 evaluation. Range (min … max): 2.553 ms … 9.490 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 2.619 ms ┊ GC (median): 0.00% Time (mean ± σ): 2.655 ms ± 259.992 μs ┊ GC (mean ± σ): 0.00% ± 0.00% ▁▆▇▄ ▁▅▅▃ ▁▇█▆▂▁ ▃▂▂▃▃██████████▆▇▅▄▃▃▃▃▃▃▃▄██████▅▄▄▃▃▃▃▂▂▂▃▂▂▂▁▂▂▂▂▂▁▂▂▂▁▂ ▄ 2.55 ms Histogram: frequency by time 2.8 ms < Memory estimate: 0 bytes, allocs estimate: 0.
# direct call of BLAS wrapper function
@benchmark LinearAlgebra.BLAS.gemm!('N', 'N', 1.0, $A, $B, 0.0, $C)BenchmarkTools.Trial: 1903 samples with 1 evaluation. Range (min … max): 2.539 ms … 7.890 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 2.565 ms ┊ GC (median): 0.00% Time (mean ± σ): 2.626 ms ± 245.462 μs ┊ GC (mean ± σ): 0.00% ± 0.00% ▇█▄▃ ▄████▆▅▄▃▃▂▂▂▂▂▂▂▂▂▂▂▂▂▂▃▄▅▆▅▅▅▅▆▄▄▃▃▃▂▂▂▂▃▂▂▂▂▁▂▂▁▂▂▂▂▂▂▂▂ ▃ 2.54 ms Histogram: frequency by time 2.84 ms < Memory estimate: 0 bytes, allocs estimate: 0.
Question (again): Can our loop beat BLAS?
Exercise: Annotate the loop in matmul_by_loop! by @turbo and @tturbo (multi-threading) and benchmark again.
using RCall
R"""
sessionInfo()
"""RObject{VecSxp}
R version 4.3.2 (2023-10-31)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Sonoma 14.4.1
Matrix products: default
BLAS: /System/Library/Frameworks/Accelerate.framework/Versions/A/Frameworks/vecLib.framework/Versions/A/libBLAS.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.11.0
locale:
[1] C
time zone: America/Los_Angeles
tzcode source: internal
attached base packages:
[1] stats graphics grDevices utils datasets methods base
loaded via a namespace (and not attached):
[1] compiler_4.3.2R"""
library(dplyr)
library(bench)
A <- $A
B <- $B
bench::mark(A %*% B) %>%
print(width = Inf)
""";┌ Warning: RCall.jl:
│ Attaching package: 'dplyr'
│
│ The following objects are masked from 'package:stats':
│
│ filter, lag
│
│ The following objects are masked from 'package:base':
│
│ intersect, setdiff, setequal, union
│
└ @ RCall /Users/huazhou/.julia/packages/RCall/dDAVd/src/io.jl:172
┌ Warning: RCall.jl: Warning: Some expressions had a GC in every iteration; so filtering is disabled.
└ @ RCall /Users/huazhou/.julia/packages/RCall/dDAVd/src/io.jl:172# A tibble: 1 x 13
expression min median `itr/sec` mem_alloc `gc/sec` n_itr n_gc
<bch:expr> <bch:tm> <bch:tm> <dbl> <bch:byt> <dbl> <int> <dbl>
1 A %*% B 125ms 126ms 7.80 30.5MB 7.80 4 4
total_time result memory time
<bch:tm> <list> <list> <list>
1 513ms <dbl [2,000 x 2,000]> <Rprofmem [1 x 3]> <bench_tm [4]>
gc
<list>
1 <tibble [4 x 3]>Re-build R from source using OpenBLAS or MKL will immediately boost linear algebra performance in R. Google build R using MKL to get started. Similarly we can build Julia using MKL.
Matlab uses MKL. Usually it’s very hard to beat Matlab in terms of linear algebra.
using MATLAB
mat"""
f = @() $A * $B;
timeit(f)
"""ArgumentError: ArgumentError: Package MATLAB not found in current path.
- Run `import Pkg; Pkg.add("MATLAB")` to install the MATLAB package.In R, the command
t(A) %*% xwill first transpose A then perform matrix multiplication, causing unnecessary memory allocation
using Random, LinearAlgebra, BenchmarkTools
Random.seed!(123)
n = 1000
A = rand(n, n)
x = rand(n);R"""
A <- $A
x <- $x
bench::mark(t(A) %*% x) %>%
print(width = Inf)
""";# A tibble: 1 x 13
expression min median `itr/sec` mem_alloc `gc/sec` n_itr n_gc
<bch:expr> <bch:tm> <bch:tm> <dbl> <bch:byt> <dbl> <int> <dbl>
1 t(A) %*% x 1.76ms 2.11ms 472. 7.64MB 95.5 178 36
total_time result memory time
<bch:tm> <list> <list> <list>
1 377ms <dbl [1,000 x 1]> <Rprofmem [2 x 3]> <bench_tm [214]>
gc
<list>
1 <tibble [214 x 3]>Julia is avoids transposing matrix whenever possible. The Transpose type only creates a view of the transpose of matrix data.
typeof(transpose(A))Transpose{Float64, Matrix{Float64}}fieldnames(typeof(transpose(A)))(:parent,)# same data in tranpose(A) and original matrix A
pointer(transpose(A).parent), pointer(A)(Ptr{Float64} @0x0000000159ba0000, Ptr{Float64} @0x0000000159ba0000)# dispatch to BLAS
# does *not* actually transpose the matrix
@benchmark transpose($A) * $xBenchmarkTools.Trial: 10000 samples with 1 evaluation. Range (min … max): 22.500 μs … 77.792 μs ┊ GC (min … max): 0.00% … 0.00% Time (median): 23.500 μs ┊ GC (median): 0.00% Time (mean ± σ): 24.330 μs ± 3.956 μs ┊ GC (mean ± σ): 0.00% ± 0.00% █▆▇▆▃▁ ▁▁▂▂▂▂▁ ▁ ▂ ████████████████▇▇▇▇███▇▇▇▆▅▅▅▆▆▅▆▅▅▄▄▃▃▁▁▁▃▁▁▁▁▁▁▁▁▁▁▁▁▆▇█ █ 22.5 μs Histogram: log(frequency) by time 48.4 μs < Memory estimate: 8.00 KiB, allocs estimate: 1.
# pre-allocate result
out = zeros(size(A, 2))
@benchmark mul!($out, transpose($A), $x)BenchmarkTools.Trial: 10000 samples with 1 evaluation. Range (min … max): 22.625 μs … 2.375 ms ┊ GC (min … max): 0.00% … 0.00% Time (median): 22.916 μs ┊ GC (median): 0.00% Time (mean ± σ): 24.165 μs ± 29.290 μs ┊ GC (mean ± σ): 0.00% ± 0.00% █▃ ▁ ▁ ████▇▅████▇▇▇▆▆▇▇▆▆▇▇▅▅▄▃▄▄▅▅▅▃▃▃▃▁▁▁▁▃▁▄▁▁▁▁▁▁▃▁▁▁▁▁▁▁▁▃▄▇ █ 22.6 μs Histogram: log(frequency) by time 49.4 μs < Memory estimate: 0 bytes, allocs estimate: 0.
# or call BLAS wrapper directly
@benchmark LinearAlgebra.BLAS.gemv!('T', 1.0, $A, $x, 0.0, $out)BenchmarkTools.Trial: 10000 samples with 1 evaluation. Range (min … max): 22.542 μs … 79.375 μs ┊ GC (min … max): 0.00% … 0.00% Time (median): 22.792 μs ┊ GC (median): 0.00% Time (mean ± σ): 23.348 μs ± 2.238 μs ┊ GC (mean ± σ): 0.00% ± 0.00% ▇█▆ ▁ ▂ ████▆▅▁▅▆▅▅▅▅▄▄▅▇█▅▅▅▇▇██████▇▅▇▇▇▆▆▅▄▅▅▅▄▅▅▆▆▇▆▆▆▆▅▅▅▆▇▇▇▇ █ 22.5 μs Histogram: log(frequency) by time 32.5 μs < Memory estimate: 0 bytes, allocs estimate: 0.
Broadcasting in Julia achieves vectorized code without creating intermediate arrays.
Suppose we want to calculate elementsize maximum of absolute values of two large arrays. In R or Matlab, the command
max(abs(X), abs(Y))will create two intermediate arrays and then one result array.
using RCall
Random.seed!(123)
X, Y = rand(1000, 1000), rand(1000, 1000)
R"""
library(dplyr)
library(bench)
bench::mark(max(abs($X), abs($Y))) %>%
print(width = Inf)
""";# A tibble: 1 x 13
expression min median `itr/sec` mem_alloc `gc/sec`
<bch:expr> <bch:tm> <bch:tm> <dbl> <bch:byt> <dbl>
1 max(abs(`#JL`$X), abs(`#JL`$Y)) 3.44ms 3.72ms 268. 15.3MB 219.
n_itr n_gc total_time result memory time
<int> <dbl> <bch:tm> <list> <list> <list>
1 65 53 242ms <dbl [1]> <Rprofmem [2 x 3]> <bench_tm [118]>
gc
<list>
1 <tibble [118 x 3]>In Julia, dot operations are fused so no intermediate arrays are created.
# no intermediate arrays created, only result array created
@benchmark max.(abs.($X), abs.($Y))BenchmarkTools.Trial: 6090 samples with 1 evaluation. Range (min … max): 254.125 μs … 3.850 ms ┊ GC (min … max): 0.00% … 77.90% Time (median): 696.666 μs ┊ GC (median): 0.00% Time (mean ± σ): 819.485 μs ± 416.072 μs ┊ GC (mean ± σ): 15.00% ± 18.68% ▃▁ ▇█▇▆▄▃▃▃▂ ▁ ▁▁▁ ▂ ██▆▄▄▁▁▁▁▁██████████▇▆▃▄▃▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▃▄▆▇███████▆██▇██████ █ 254 μs Histogram: log(frequency) by time 2.32 ms < Memory estimate: 7.63 MiB, allocs estimate: 2.
Pre-allocating result array gets rid of memory allocation at all.
# no memory allocation at all!
Z = zeros(size(X)) # zero matrix of same size as X
@benchmark $Z .= max.(abs.($X), abs.($Y)) # .= (vs =) is important!BenchmarkTools.Trial: 10000 samples with 1 evaluation. Range (min … max): 213.292 μs … 415.042 μs ┊ GC (min … max): 0.00% … 0.00% Time (median): 222.750 μs ┊ GC (median): 0.00% Time (mean ± σ): 230.535 μs ± 14.789 μs ┊ GC (mean ± σ): 0.00% ± 0.00% ▆▃▃▆ █▃ ▂▄▄█████▅▄▄▃▃▂▂▂▂▂▂▁▁▁▁▁▁▁▃██▃▄▄▃▂▂▂▂▂▂▂▂▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁▁ ▂ 213 μs Histogram: frequency by time 276 μs < Memory estimate: 0 bytes, allocs estimate: 0.
Exercise: Annotate the broadcasting by @turbo and @tturbo (multi-threading) and benchmark again.
View avoids creating extra copy of matrix data.
Random.seed!(123)
A = randn(1000, 1000)
# sum entries in a sub-matrix
@benchmark sum($A[1:2:500, 1:2:500])BenchmarkTools.Trial: 10000 samples with 1 evaluation. Range (min … max): 45.292 μs … 3.645 ms ┊ GC (min … max): 0.00% … 97.74% Time (median): 68.708 μs ┊ GC (median): 0.00% Time (mean ± σ): 80.365 μs ± 156.568 μs ┊ GC (mean ± σ): 14.19% ± 7.15% █ ▄▃▂▁▂▂▃▃▂▂▂▂▂▂▂▂▂▂▂▂▁▁▂▂▇█▇▄▅▇█▆▆▅▅▅▅▅▄▄▃▃▃▃▃▃▂▂▂▂▂▂▂▂▂▂▂▂▂▂ ▃ 45.3 μs Histogram: frequency by time 91.6 μs < Memory estimate: 488.42 KiB, allocs estimate: 2.
# view avoids creating a separate sub-matrix
@benchmark sum(@view $A[1:2:500, 1:2:500])BenchmarkTools.Trial: 10000 samples with 1 evaluation. Range (min … max): 50.708 μs … 106.583 μs ┊ GC (min … max): 0.00% … 0.00% Time (median): 55.042 μs ┊ GC (median): 0.00% Time (mean ± σ): 56.216 μs ± 3.059 μs ┊ GC (mean ± σ): 0.00% ± 0.00% █▁ ▂▂▁▁▁▁ ▁ ▁ ▇▄▁▃▁▁▃▁▁▁▃▁▁▄██▇▇▆▆▆▅▆▆▆██████▇▇▆▆▆▆██████████▆▆▆▆▇▇▆▆▆▆▇▅▆ █ 50.7 μs Histogram: log(frequency) by time 68.9 μs < Memory estimate: 0 bytes, allocs estimate: 0.
The @views macro, which can be useful in some operations.
@benchmark @views sum($A[1:2:500, 1:2:500])BenchmarkTools.Trial: 10000 samples with 1 evaluation. Range (min … max): 54.917 μs … 106.917 μs ┊ GC (min … max): 0.00% … 0.00% Time (median): 55.042 μs ┊ GC (median): 0.00% Time (mean ± σ): 56.096 μs ± 2.914 μs ┊ GC (mean ± σ): 0.00% ± 0.00% █▄ ▂▁ ▁ ▁ ▁ ██▅▄▇▆▄▃▄▅▃▄▃▄▃██▇███▇█▆▆▅▅▃▄▅▄▆▆▆▆▇▆▇▇▅▆█▇▇▆▇▅▄▄▄▄▄▅▅▅▅▅▅▅▅ █ 54.9 μs Histogram: log(frequency) by time 68.2 μs < Memory estimate: 0 bytes, allocs estimate: 0.